Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000206: Integer partitions ⟶ ℤ
Values
[1,0,1,0] => [1] => [1] => 0
[1,0,1,0,1,0] => [2,1] => [1,1,1] => 0
[1,0,1,1,0,0] => [1,1] => [2] => 0
[1,1,0,0,1,0] => [2] => [1,1] => 0
[1,1,0,1,0,0] => [1] => [1] => 0
[1,0,1,0,1,0,1,0] => [3,2,1] => [3,1,1,1] => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [4,1] => 0
[1,0,1,1,0,0,1,0] => [3,1,1] => [3,2] => 1
[1,0,1,1,0,1,0,0] => [2,1,1] => [2,1,1] => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [2,1] => 0
[1,1,0,0,1,0,1,0] => [3,2] => [3,1,1] => 0
[1,1,0,0,1,1,0,0] => [2,2] => [4] => 0
[1,1,0,1,0,0,1,0] => [3,1] => [3,1] => 0
[1,1,0,1,0,1,0,0] => [2,1] => [1,1,1] => 0
[1,1,0,1,1,0,0,0] => [1,1] => [2] => 0
[1,1,1,0,0,0,1,0] => [3] => [3] => 0
[1,1,1,0,0,1,0,0] => [2] => [1,1] => 0
[1,1,1,0,1,0,0,0] => [1] => [1] => 0
[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [4,3,1] => 53
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [4,1,1,1] => 0
[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [6,2] => 15
[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [2,1,1,1,1,1,1] => 0
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [3,2,1,1] => 8
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [4,2] => 3
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [2,1,1,1,1,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [3,2,1] => 6
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [2,1,1,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [2,2] => 0
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [6,1,1] => 0
[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [4,1,1,1,1] => 0
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [4,3] => 11
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [4,1,1] => 0
[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [3,1,1,1,1,1] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [6,1] => 0
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [1,1,1,1,1,1,1] => 0
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [3,1,1,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [4,1] => 0
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [2,1,1,1,1] => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [3,2] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [2,1,1] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [2,1] => 0
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [3,1,1,1,1] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [6] => 0
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [1,1,1,1,1,1] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [3,1,1] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [4] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [1,1,1,1,1] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [3,1] => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,1,1] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [2] => 0
[1,1,1,1,0,0,0,0,1,0] => [4] => [1,1,1,1] => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [3] => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,1] => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => [1] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [4,2,1,1] => 31
[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [3,2,1,1,1] => 11
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [4,2,1] => 16
[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => [2,2,1,1,1,1] => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [3,2,2] => 8
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [2,2,1,1] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [2,2,1] => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [4,4] => 14
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => [4,3,1] => 53
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [4,1,1,1] => 0
[1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => [6,2] => 15
[1,1,0,1,1,0,1,0,0,1,0,0] => [4,2,1,1] => [2,1,1,1,1,1,1] => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [3,2,1,1] => 8
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [4,2] => 3
[1,1,0,1,1,1,0,0,0,0,1,0] => [5,1,1,1] => [5,2,1] => 43
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [2,1,1,1,1,1] => 0
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [3,2,1] => 6
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [2,1,1,1] => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [2,2] => 0
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [6,1,1] => 0
[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => [4,1,1,1,1] => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [4,3] => 11
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [4,1,1] => 0
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => [3,1,1,1,1,1] => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [6,1] => 0
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,1] => [5,1,1,1] => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [1,1,1,1,1,1,1] => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [3,1,1,1] => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [4,1] => 0
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [5,2] => 7
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [2,1,1,1,1] => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [3,2] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [2,1,1] => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [2,1] => 0
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => [8] => 0
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => [5,3] => 31
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [3,1,1,1,1] => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [6] => 0
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [5,1,1] => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [1,1,1,1,1,1] => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [3,1,1] => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [4] => 0
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [5,1] => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [1,1,1,1,1] => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [3,1] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,1,1] => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [2] => 0
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Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many integer compositions $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight..
Each value in this statistic is greater than or equal to corresponding value in St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight..
Given $\lambda$ count how many integer compositions $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight..
Each value in this statistic is greater than or equal to corresponding value in St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight..
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
Glaisher-Franklin inverse
Description
The Glaisher-Franklin bijection on integer partitions.
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].
This map sends the number of distinct repeated part sizes to the number of distinct even part sizes, see [1, 3.3.1].
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